Types of Homology Theories  

In General > s.a. homology theory.
* Different coefficients: Homology groups using \(\mathbb Q\) or \(\mathbb R\) instead of \(\mathbb Z\) contain less information about the topological space; They cannot have any torsion subgroup.
* Simplicial homology: Defined for polyhedra, in a simple way; For a general space X, one triangulates X with some polyhedron K, and shows that \(H_q(K)\) does not depend on K.
* Singular simplicial homology: A generalization of the simplicial one; Instead of triangulating the whole space, one just looks at continuous (but possibly "singular", non-invertible) maps from simplices into it; More formally, one constructs a functor to the category of chain complexes, and from there another one to the Abelian groups.
* Singular cubical homology: Same as singular simplicial homology, but with maps of cubes rather than simplices.
* Čech homology: Defined using simplices, becomes independent of simplices through some limiting procedure.
@ Singular simplicial homology: Eilenberg AM(44) [original paper, topologically-invariant manner].
@ K-homology: Douglas 80.
@ Morse homology: Banyaga & Hurtubise 04.
@ Floer homology: Zois ht/05 [non-commutative]; > s.a. 4-manifolds; non-commutative field theory.
@ Knot homology: Gukov & Saberi in(14)-a1211-ln [and quantum curves]; Gorsky a1304 [colored homology of knots]; Nawata & Oblomkov a1510-proc.

Examples > s.a. causal sets [stable homology]; graph invariants; networks [persistent homology].
* Decomposition: In general, we can write Hq(K) = GqTq, where Gq is free and its rank gives the number of (q+1)-dimensional holes in K, the q-th Betti number, while the torsion subgrop Tq tells how K is twisted.
* For manifolds: The 0-th one is H0(M) = \(\mathbb Z\) ⊕ \(\mathbb Z\) ⊕ ... ⊕ \(\mathbb Z\), with as many \(\mathbb Z\)s as connected components in M; If M is a manifold of dimension n, then Hq(M) = 0, for all q > n (this is not true for the πq).
* For Rn: Hm(\(\mathbb R\)n, \(\mathbb R\)n \{0}) = δmn\(\mathbb Z\), for m > 1.
* For spheres: H1(S1) = \(\mathbb Z\) (> s.a. Hurewicz Theorem).

Relative Homology
* Idea: We take a subpolyhedron L of K and consider like the identity anything in K belonging to L; Define the following chain complex,

Cp(K; L):= Cp(K)/Cp(L) ,   p > 0 ,   and
\(\bar\partial\)p: Cp(K; L) → Cp−1(K; L)   by   \(\bar\partial\)p(cp + Cp(L)):= ∂p cp + Cp−1(L) ,   cpCp(K) .

* Motivation: The usefulness lies in the fact that Hp(K) → Hp(L) → Hp(K; L) is an exact sequence.

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